Green and MacCullagh

The correct boundary-conditions were thus obtained. It became easier to discuss the reflexion and refraction of an incident wave by the procedure of Fresnel and Cauchy.

Green found that if the vibration of the aethereal molecules is executed at right angles to the plane of incidence, the intensity of the reflected light obeys Fresnel’s sine-law, provided the rigidity n is the same for all media, but the inertia ρ varies from one medium to another.

Fresnel’s hypotheses was that:

• the vibrations are executed at right angles to the plane of polarization, and
• the optical differences between media are due to the different densities of aether within them.

Since the sine-law is true for light polarized in the plane of incidence, Green’s conclusion confirmed these hypotheses.

It now remained for Green to discuss the case in which the incident light is polarized at right angles to the plane of incidence, so that the motion of the aethereal particles is parallel to the intersection of the plane of incidence with the front.

In this case, it is impossible to satisfy all the six boundary-conditions without assuming that longitudinal vibrations are generated by the act of reflexion.

Taking the plane of incidence to be the plane of yz, and the interface to be the plane of xy, the incident wave may be represented by the equations

where, if i deote the angle of incidence, we have

There will be a transverse reflected wave,

and a transverse refracted wave,

where, since the velocity of transverse waves in the second medium is

there will also be a longitudinal reflected wave,

where λ, is determined by the equation

and a longitudinal refracted wave,

where λ1, is determined by

Substituting these values for the displacement in the boundary-conditions which have been already formulated, we obtain the equations which determine the intensities of the reflected and refracted waves; in particular, it appears that the amplitude of the reflected transverse wave is given by the equation

If the elastic constants of the media are such that the velocities of propagation of the longitudinal waves are of the same order of magnitude as those of the transverse waves, the direction-cosines of the longitudinal reflected and refracted rays will in general have real values, and these rays will carry away some of the energy which is brought to the interface by the incident wave.

Green avoided this difficulty by adopting Fresnel’s suggestion that the resistance of the aether to compression may be very large in comparison with the resistance to distortion, as is actually the case with such substances as jelly and caoutchouc: in this case the longitudinal waves are degraded in much the same way as the transverse refracted ray is degraded when there is total reflexion, and so do not carry away energy. Making this supposition, so that k1 and k2 are very large, the quantities λ and λ1, have the values m

…

and we have

Thus it

…

denote the modulus of B/A, we have

This expression represents the ratio of the intensity of the transverse reflected wave to that the incident wave.

It does not agree with Fresnel’s tangent-formula: and both on this account and also because (as we shall see) this theory of reflexion does not harmonize well with the clastic-solid theory of crystal-optics, it must be concluded that the vibrations of a Greenian solid do not furnish an exact parallel to the vibrations which constitute light.

The success of Green’s investigation from the standpoint of dynamics, set off by its failure in the details last mentioned, stimulated MacCullagh to fresh exertions.

At length he succeeded in placing his own theory, which had all along been free from reproach so far as agreement with optical experiments was concerned, on a sound dynamical basis; thereby effecting that reconciliation of the theories of Light and Dynamics which had been the dream of every physicist since the days of Descartes.

The central feature of MacCullagh’s investigation,[27] which was presented to the Royal Irish Academy in 1839, is the introduction of a new type of elastic solid. He had, in fact, concluded from Green’s results that it was impossible to explain optical phenomena satisfactorily by comparing the aether to an elastic solid of the ordinary type, which resists compression, and distortion; and he saw that the only hope of the situation was to devise a medium which should be as strictly conformable to dynamical laws as Green’s elastic solid, and yet should have its properties specially designed to fulfil the requirements of the theory of light. Such a medium he now described.

If as before we denote by e the vector displacement of a point of the medium from its equilibrium position, it is well known that the vector curl e denotes twice the rotation of the part of the solid in the neighbourhood of the point (x, y, z) from its equilibrium orientation. In an ordinary elastic solid, the potential energy of strain depends only on the change of size and shape of the volume-elements; on their compression and distortion, in fact. For MacCullagh’s new medium, on the other hand, the potential energy depends only on the rotation of the volume-elements.

Since the medium is not supposed to be in a state of stress in its undisturbed condition, the potential energy per unit volume must be a quadratic function of the derivates of e; so that in an isotropic medium this quantity φ must be formed from the only in variant which depends solely on the rotation and is quadratic in the derivates, that is from (curl e)2; thus we may write

The equation of motion is now to be determined, as in the case of Green’s aether, from the variational equation

the result is

From this equation, if div e is initially zero it will always be zero: we shall suppose this to be the case, so that no longitudinal waves exist at any time in the medium. One of the greatest difficulties which beset elastic-solid theories is thus completely removed.

The equation of motion may now be written

which shows that transverse waves are propagated with velocity

From the variational equation we may also determine the boundary-conditions which must be satisfied at the interface between two media; these are, that the three components of e are to be continuous across the interface, and that the two components of curl e parallel to the interface are also to be continuous across it. One of these five conditions, namely, the continuity of the normal component of e, is really dependent on the other four; for if we take the axis of x normal to the interface, the equation of motion gives

and as the quantities ρ, (μ curl e)z and (μ curl e)y, are continuous across the interface, the continuity of ∂2ex/∂t2 follows. Thus the only independent boundary-conditions in MacCullagh’s theory are the continuity of the tangential components of e and of μ curl e.[28] It is easily seen that these are equivalent to the boundary-conditions used in MacCullagh’s earlier paper, namely, the equation of vis viva and the continuity of the three components of e: and thus the “rotationally elastic” aether of this memoir furnishes a dynamical foundation for the memoir of 1837.

The extension to crystalline media is made by assuming the potential energy per unit volume to have, when referred to the principal axes, the form

where A, B, C denote three constants which determine the optical behaviour of the medium: it is readily seen that the wave-surface is Fresnel’s, and that the plane of polarization contains the displacement, and is at right angles to the rotation.

MacCullagh’s work was regarded with doubt by his own and the succeeding generation of mathematical physicists. It was not appreciated until FitzGerald drew attention to it 40 years later.

But MacCullagh really solved the problem of devising a medium whose vibrations, calculated in accordance with the correct laws of dynamics, should have the same properties as the vibrations of light.

The hesitation which was felt in accepting the rotationally elastic aether arose mainly from the want of any readily conceived example of a body endowed with such a property. This difficulty was removed in 1889 by Sir William Thomson (Lord Kelvin), who designed mechanical models possessed of rotational elasticity.

Suppose, for example,[29] that a structure is. formed of spheres, each sphere being in the centre of the tetrahedron formed by its four nearest neighbours. Let each sphere be joined to these four neighbours by rigid bars, which have spherical caps at their ends so as to slide freely on the spheres.

Such a structure would, for small deformations, behave like an incompressible perfect fluid. Now attach to each bar a. pair of gyroscopically-mounted flywheels, rotating with equal and opposite angular velocities, and having their axes in the line of the bar: a bar thus equipped will require a couple to hold it at rest in any position inclined to its original position, and the structure as a whole will possess that kind of quasi-elasticity which was first imagined by MacCullagh.

This particular representation is not perfect, since a system of forces would be required to hold the model in equilibrium if it were irrotationally distorted. Lord Kelvin subsequently invented another structure free from this defect.[30]